Question 1191373
Here's how to calculate the Poisson probabilities:

**Understanding the Poisson Distribution**

The Poisson distribution is used to model the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known average rate and independently of the time since the last event. The formula is:

P(x) = (e^(-λ) * λ^x) / x!

Where:

* P(x) is the probability of x events occurring
* e is the base of the natural logarithm (approximately 2.71828)
* λ is the average number of events (arrivals) per interval (10 in this case)
* x is the number of events we're interested in
* x! is the factorial of x

**c) Fewer than five arrivals (x < 5):**

We need to calculate P(0) + P(1) + P(2) + P(3) + P(4).

* P(0) = (e^-10 * 10^0) / 0! ≈ 0.000045
* P(1) = (e^-10 * 10^1) / 1! ≈ 0.000454
* P(2) = (e^-10 * 10^2) / 2! ≈ 0.002270
* P(3) = (e^-10 * 10^3) / 3! ≈ 0.007567
* P(4) = (e^-10 * 10^4) / 4! ≈ 0.018918

P(x < 5) = P(0) + P(1) + P(2) + P(3) + P(4) ≈ 0.000045 + 0.000454 + 0.002270 + 0.007567 + 0.018918 ≈ 0.029254

Rounded to 4 decimal places, the probability is 0.0293.

**d) At least 11 arrivals (x ≥ 11):**

It's easier to calculate the complement and subtract from 1.  That is:

P(x ≥ 11) = 1 - P(x < 11) = 1 - [P(0) + P(1) + ... + P(10)]

Calculating each individual probability from P(0) to P(10) and summing them can be tedious.  It's best to use a calculator or statistical software that has built-in Poisson cumulative distribution functions (CDF).
Using a calculator or software, you'll find that P(x < 11) ≈ 0.5830.

Therefore, P(x ≥ 11) = 1 - 0.5830 = 0.4170

Rounded to 4 decimal places, the probability is 0.4170.